In previous papers the autor has defined a quasi-order $\preceq$ on the class of groups (the``largeness'' ordering). One can then define the {\it height} of group, and also define what it means for a group to satisfy max-$\preceq$ or min-$\preceq$. A natural question is whether the finiteness conditions max-$\preceq$, min-$\preceq$, ``having finite height'' are extension closed. It is shown here that the answer is ``no'' for all three properties: there is a group which is a split extension of one group of height 1 by another group of height 1, and which does not satisfy max-$\preceq$ or min-$\preceq$.